A recent preprint of Ohashi (arXiv:2604.18074) constructs, in every characteristic p>5, three explicit families of curves of genus 4, 5 and 6 whose Jacobians are completely decomposable, and reduces the question of whether a member is superspecial to whethe…
Three general conics in the plane admit 184 complex circles tangent to all three, and Breiding, Lindberg, Ong and Sommer conjectured that at most 136 of them can be real. Brysiewicz has recently refuted that conjecture by exhibiting a triple of conics with…
Let K be a field of characteristic zero and let q₁,…,qₘ be quadratic forms in n variables over K. In his work on uniqueness of powers-of-forms decompositions, Taveira Blomenhofer writes β₂(n) for the largest m for which some such family admits no nonzero al…
Let r ≥ 2, let q ≡ 1 (mod r) be a prime power with p ∤ (r²-1), put m=(q-1)/r, and let k ∈ 𝔽_q^(*) be such that -k is not an r-th power. Asgarli, Ghioca and Yip proved that the plane curve -(ky+z)xᵐ+zyᵐ+kyzᵐ=0 is smooth and nontrivially blocking as soon as…
Let X: yⁿ=f(x) be a superelliptic curve over 𝔽_q with f separable of degree d ≥ 3 splitting into distinct linear factors and with n | q-1. Say that the pair (n,f) lies in the split locus when every value of f outside its roots is an n-th power in 𝔽_q^(*),…
A general pair of twisted cubic curves in ℙ³(ℂ) has ten common secant lines. For a real pair each real common secant is totally, partially or minimally real according to whether it meets the two curves in two real points each, in two real points and a conju…
The Jacobian conjecture was refuted in dimension three in July 2026, and a general construction of Gao — sweeping tangent direction fields on parametrised hypersurfaces — has since produced Keller counterexamples in every dimension greater than two. Four of…
A recent chain realization theorem of Castañeda, Honorato and Valenzuela-Henríquez turns a polynomial Keller map F of ℝⁿ with a multi-point fiber into a polynomial vector field on ℝ^N, N=Σᵢmax(deg Fᵢ,2)-n, whose Jacobian has spectrum {-1} everywhere and who…
